I followed the analysis of this recently published paper Jan Meyer et al 2022 Class. Quantum Grav. 39 135001 to calculate the birefringence noise in the CTN experiment. Interestingly, the contribution from birefringence noise after my first attempt at this calculation looks very close to what we were calculating as coating thermorefractive noise before. If this were true, our experiment would have seen it much before. In fact, we wouldn't have seen thermooptic cancellation as Tara experimentally verified here. So something is missing
What is birefringence noise?
After going through some literature and reading properly Meyer et al, I have the following understanding of the birefringence noise (and why it is called so).
 The temperature fluctuations cause length fluctuations in the coating layers (through the coefficient of thermal expansion)
 The length fluctuations cause stress fluctuations in the coating layers (through Young's modulus Y).
 The stress fluctuations get converted into refractive index fluctuations through the photoelastic effect (through photoelastic tensor)
 The photoelastic tensor for cubic crystals like GaAs has 3 independent values, P11, P12 and P44. (Section 12: Table 1. ElastoOptic Coefficients: Cubic Crystals (43m, 432, m3m))
 This induces refractive index fluctuations that are different for the fast and slow axes. The difference between the two fluctuations causes a phase shift in reflected light from each layer. That's why this can be birefringence noise. In an unstressed isotropic material, this pathway should not exist.
Is this different from thermorefractive noise?
This is a question I am still not sure how to answer. My understanding is that the common mode change in refractive indices of both axes drives the thermorefractive noise. This means I should be able to derive the coefficient of thermorefraction using the same formalism.
Calculation:
Both thermorefractive noise and thermophotoelastic noise show up as dn/dT terms in the thermooptic noise summation, just through different physical processes. This could mean that experimentally measured coefficients of thermorefraction already include birefringent contribution if any. In my calculations for the plots presented here, I got the following values of the two coefficients:
Coefficient of thermorefraction (Effective for coating): 8.289e05
Coefficient of thermophotoelastic effect (Effective for coating, using Eq.11 of Meyer et al.): 8.290e05
It was very surprising to me to see that both these coefficients came out to be within 1% of each other.
Because of this, when we add the noise sources coherently (since they are all driven by the same thermal fluctuations), the thermooptic cancellation that we have experimentally proved does not work anymore. So something must be wrong with my calculation.
Possible explanations:
 Calculaiton error in my code. I'll double check tomorrow.
 Somehow the thermorefractive noise already takes into account the birefringent noise, through the coefficient of thermorefraciton that we use as seed in our thermorefractive noise calculation. This would explain how the witnessed themooptic cancellation was achieved.
 Meyer et al. is calculating birefringent noise for the substrate. Maybe the tensorial calculations are different for coatings.
